Materials Thermal Exp Coeff Metals

Thermal expansion and the cost function

The thermal expansion coefficient of a metal measures how much its length changes per degree of temperature change. For iron it is about 12, for copper about 17, and for aluminum about 24, all in units of 10^-6 per kelvin. Aluminum expands about twice as much as iron, and that ratio of roughly 2 is close to the golden ratio raised to the power 1.4. The special alloy Invar expands almost not at all, around 1 x 10^-6 per kelvin, which is why it appears in precision instruments.

In Recognition Science, the framework models a ledger, a discrete record of events, and assigns a forced cost to each recognition. The cost function J(x) = (x + 1/x)/2 - 1 has a proved uniqueness theorem in the machine-checked library of formal theorems. The library defines domainCost as J applied to the ratio of two quantities, m and e. For thermal expansion, the research note suggests m and e might stand for measured and expected coefficients, but the Lean code itself never defines them that way. The definition of domainCost is generic, shared verbatim with 2383 sibling modules.

What the module actually proves are three general facts about this cost function. First, the cost is zero when the two quantities are equal. Second, the cost is never negative for positive inputs. Third, the number phi - 3/2 is positive, where phi is the golden ratio. These are true statements about J applied to any positive ratio; they say nothing specific about iron, copper, or Invar.

The gap matters. A theorem about metal expansion would need a definition of m and e in the metal's own terms, such as a measured coefficient and a predicted one from the phi-ladder. No such definition exists in the module. The docstring is explicit: the paragraph about CTE values clustering at J(phi)^n is a research note recording where the idea was meant to go, not a result. The honest summary is that the framework proves the cost function's basic hygiene, and the thermal expansion story remains a target for future work.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove any specific thermal expansion coefficient for iron, copper, aluminum, or Invar. The ratio Al/Fe being close to 2 is an empirical observation, not a derived theorem. No definition connects m and e to measured and expected CTE values in the Lean code.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/ThermalExpCoeffMetals.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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